{"id":436,"date":"2012-06-22T22:36:35","date_gmt":"2012-06-22T13:36:35","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=436"},"modified":"2021-09-30T08:45:35","modified_gmt":"2021-09-29T23:45:35","slug":"kbr201206","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201206\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2012\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(2\\) \u3064\u306e\u53cc\u66f2\u7dda \\(C : \\ x^2 -y^2 = 1\\) , \\(H : \\ x^2 -y^2 = -1\\) \u3092\u8003\u3048\u308b. \u53cc\u66f2\u7dda \\(H\\) \u4e0a\u306e\u70b9 \\(P (s,t)\\) \u306b\u5bfe\u3057\u3066, \u65b9\u7a0b\u5f0f \\(sx -ty = 1\\) \u3067\u5b9a\u307e\u308b\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u76f4\u7dda \\(\\ell\\) \u306f\u70b9 \\(P\\) \u3092\u901a\u3089\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u76f4\u7dda \\(\\ell\\) \u3068\u53cc\u66f2\u7dda \\(C\\) \u306f\u76f8\u7570\u306a\u308b \\(2\\) \u70b9 \\(Q , R\\) \u3067\u4ea4\u308f\u308b\u3053\u3068\u3092\u793a\u3057, \\(\\triangle PQR\\) \u306e\u91cd\u5fc3 \\(G\\) \u306e\u5ea7\u6a19\u3092 \\(s , t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u306b\u304a\u3051\u308b \\(3\\) \u70b9 \\(G , Q , R\\) \u306b\u5bfe\u3057\u3066, \\(\\triangle GQR\\) \u306e\u9762\u7a4d\u306f\u70b9 \\(P (s,t)\\) \u306e\u4f4d\u7f6e\u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(P\\) \u306f \\(H\\) \u4e0a\u306e\u70b9\u306a\u306e\u3067\r\n\\[\r\ns^2 +t^2 = -1 \\quad ... [1]\r\n\\]\r\n\\(\\ell\\) \u304c \\(P\\) \u3092\u901a\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\r\ns \\cdot s -t \\cdot t = s^2-t^2 = 1\r\n\\]\r\n\u3053\u308c\u306f [1] \u306b\u77db\u76fe\u3059\u308b\u306e\u3067, \\(\\ell\\) \u306f \\(P\\) \u3092\u901a\u3089\u306a\u3044.<\/p>\r\n

    (2)<\/strong><\/p>\r\n

    \\(t \\neq 0\\) \u306a\u306e\u3067, \\(\\ell\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny = \\dfrac{sx-1}{t}\r\n\\]\r\n\u3053\u308c\u3092 \\(C\\) \u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nt^2 x^2 -(sx-1)^2 & = t^2 \\\\\r\n(t^2-s^2) x^2 +2sx -t^2-1 & = 0 \\\\\r\n\\text{\u2234} \\quad x^2 +2sx -(t^2+1) & = 0 \\quad ... [2]\r\n\\end{align}\\]\r\n\u3053\u308c\u306e\u5224\u5225\u5f0f\u3092 \\(D\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\dfrac{D}{4} =s^2+t^2+1 \\gt 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(C\\) \u3068 \\(\\ell\\) \u306f\u7570\u306a\u308b \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b.
    \r\n\u3055\u3089\u306b, [2] \u3092\u89e3\u3044\u3066\r\n\\[\r\nx = -s \\pm \\sqrt{s^2+t^2+1} = -s \\pm \\sqrt{2} t\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\ny & = \\dfrac{s \\left( -s \\pm \\sqrt{2} t \\right) -1}{t} \\\\\r\n& = \\dfrac{-s^2-1 \\pm \\sqrt{2} st}{t} \\\\\r\n& = -t \\pm \\sqrt{2} s\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nQ \\left( -s +\\sqrt{2} t , -t+\\sqrt{2} s \\right) , \\ R \\left( -s -\\sqrt{2} t , t-\\sqrt{2} -s \\right)\r\n\\]\r\n\u3068\u8868\u305b\u308b\u306e\u3067, \u6c42\u3081\u308b\u91cd\u5fc3 \\(G\\) \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( -\\dfrac{s}{3} , -\\dfrac{t}{3} \\right)}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\overrightarrow{PQ} & = \\left( -2s -\\sqrt{2} t , -2t -\\sqrt{2} s \\right) , \\\\\r\n\\overrightarrow{PR} & = \\left( -2s +\\sqrt{2} t , -2t +\\sqrt{2} s \\right)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\triangle GQR & = \\dfrac{1}{3} \\triangle PQR \\\\\r\n& =\\dfrac{1}{6} \\left| \\left( -2s -\\sqrt{2} t \\right) \\left( -2t +\\sqrt{2} s \\right) -\\left( -2t -\\sqrt{2} s \\right) \\left( -2s +\\sqrt{2} t \\right) \\right| \\\\\r\n& =\\dfrac{1}{6} \\left| 4 \\sqrt{2} (t^2-s^2) \\right| \\\\\r\n& = \\dfrac{2 \\sqrt{2}}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u4e00\u5b9a\u3067\u3042\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u3064\u306e\u53cc\u66f2\u7dda \\(C : \\ x^2 -y^2 = 1\\) , \\(H : \\ x^2 -y^2 = -1\\) \u3092\u8003\u3048\u308b. \u53cc\u66f2\u7dda \\(H\\) \u4e0a\u306e\u70b9 \\(P (s,t)\\) \u306b\u5bfe\u3057\u3066, \u65b9\u7a0b\u5f0f \\(sx -t … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[65],"tags":[144,68],"class_list":["post-436","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2012","tag-tsukuba_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=436"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/436\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=436"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=436"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}